**6. Results and Discussion**

The section discusses the numerical outcomes obtained on the model in consideration. The parameters involved in the results are *φ*, *N<sup>π</sup>*, *<sup>υ</sup>*<sup>∗</sup>, *B<sup>π</sup>*, *<sup>τ</sup>*<sup>∗</sup>, *ς*<sup>∗</sup>, *F<sup>π</sup>*, S, *Λπ*, *R<sup>π</sup>*, and *BΓ*. Figures 2–21 display the physical behavior of the mentioned parameters regarding energy, entropy formation, and velocity on the nondimensional entities of the model. Results for non-Newtonian Al2O3-EO and Cu-EO P-ENFs were obtained. Temperature differences and coefficient of skin fraction are detailed in Table 5. The values used for the parameters are *φ* = 0.18, *Nπ* = 0.3, *υ*<sup>∗</sup> = 0.2, *Bπ* = 0.3, *τ*<sup>∗</sup> = 1.0, *ς*∗ = 0.2, *Fπ* = 0.6, S = 0.5, *Λπ* = 0.3, *Rπ* = 5, and *BΓ* = 5. The power of Al2O3-EO and Cu-EO were decided with the fractional size of nanoparticles used in the working fluid. Flow stability of nanoparticles decreased when nanoparticles had a higher amount of fractional range. Al2O3-EO was favored more by fractional improvement than Cu-EO nanofluid. Figure 2 shows a lower flow of Cu-EO nanofluid than Al2O3-EO. As Al2O3 has a high heat transfer property, its primary purpose is to combine with EO. When the fractional volume of both fluids flows increases, thermal distribution transported to the domain from the surface is high, as shown in Figure 3. The increasing fractional volume also resulted in enhancing the fluctuations of the system entropy. Figure 4 shows the leading fluctuations of Cu-EO nanofluid, which settled down midway and increased further towards Al2O3-EO nanofluid. The thermal radiation parameter (*<sup>N</sup>π*) models the radiation procedure used in enhancing the entropy rate and heat regarding induced temperature, as shown in Figures 5 and 6. Radiations had a negligible effect on the entropy variations caused by the prominent influence of flow conditions. Cu-EO had more control than the Al2O3-EO nanofluid. Regarding heat capability of Al2O3 and Cu-EO nanofluids, there was a dominant effect on entropy and thermal aspects of individual variations in *<sup>υ</sup>*<sup>∗</sup>, i.e., thermal conductivity. Figures 7 and 8 represent these effects. When the variation parameter tries to increase the ranges of entropy and heat, the nominal impact of *υ*<sup>∗</sup> is proved by close variations of entropy and thin layers of heat. In both behaviors of parameters, Al2O3-EO underestimated the Cu-EO nanofluid. Figures 9 and 10 clearly show the increment in convective heat on thermal as well as entropy states from lower surfaces in the domain. Parametric values of Biot number represent the ordinary heating procedure *B<sup>π</sup>*. An increase in *Bπ* resulted in enhancing the thermal state in the flow domain, but this only had a negligible impact on the entropy generation. The entropy profile is smaller than the thermal boundary layer, which proves the above statement. According to the study, Al2O3-EO is better than Cu-EO nanofluid. Figures 11–13 demonstrate the impact on the power, entropy, and velocity distributions of Prandtl–Eyring nanofluid *<sup>τ</sup>*<sup>∗</sup>. Figure 11 shows the speed ( *f* ) corresponding to *<sup>τ</sup>*<sup>∗</sup>. The velocity of both fluids increased with amplification in *<sup>τ</sup>*<sup>∗</sup>. However, the velocity of Al2O3- EO velocity was more incredible as compared to Cu-EO. Figure 12 shows the temperature curve concerning the Prandtl–Eyring parameter *<sup>τ</sup>*<sup>∗</sup>. An increment in *τ*<sup>∗</sup> resulted in reducing the temperatures of both fluids. However, the temperature profile of Cu-EO nanofluid is more critical than Al2O3-EO nanofluid. Figure 13 represents the entropy fluctuation of P-ENF caused by *<sup>τ</sup>*<sup>∗</sup>. An increase of *τ*<sup>∗</sup> resulted in lowering the entropy formation. The lower value of entropy of the Al2O3-EO fluid was used to represent Cu-EO nanofluid when both nanofluids were at the end of the graph. *τ*<sup>∗</sup> is strongly related to the profile of P-ENF. However, an increase of *τ*<sup>∗</sup> resulted in decreasing the entropy and temperature. Figures 14–16 illustrate the efficacy of the Prandtl–Eyring parameter *ς*∗ on the profiles of temperature, velocity, and entropy formation. The velocity change regarding *ς*∗ was displayed in Figure 16. A decrease in the velocity profile was the result of an increment in Cu-EO while increasing Al2O3-EO and a high rate in *ς*<sup>∗</sup>. Figure 15 shows the fluctuations in the profile of temperature concerning *ς*<sup>∗</sup>. The temperature grows as *ς*∗ is increased, and Cu-EO obtains a quick temperature. Figure 16 highlights the difference in entropy caused by the Prandtl–Eyring parameter *ς*<sup>∗</sup>. An increment in entropy is obtained with increasing *ς*<sup>∗</sup>. Results obtained from modifying the slip conditions on the nature of the flow, heat, and generation of entropy, respectively, are shown in Figures 17–19. Viscous behavior was focused on the flow conditions in the combinations of the Prandtle-Eyring fluid. Variations in velocity, entropy formation, and thermal distributions have an essential role in slip conditions. The situation for fluidity becomes difficult when slip conditions of Prandtl–Eyring fluid flow are increased. Fluidity was reduced for Cu-EO than Al2O3-EO P-ENF. Such hierarchy mainly occurs in thermal distributions, i.e., Cu-EO has a higher thermal state than Al2O3-EO nanofluid, as depicted in Figure 18. Greater values of slip parameter Λ*π* resulted in decreasing the entropy generation. It was caused by slip flow, which acted opposite to entropy generation, as shown in Figure 19. Figure 20 shows the performed estimations for *Fπ* = 0.6, 1.6, and 2.6; meanwhile, parametric values of *ς*∗ are 0.2, 0.4, and 0.6. An increment in the material parameter resulted in enhancing the coefficient of skin friction. Flow velocity was decreased due to an increase in skin friction as resistance in fluid increased. In Figure 21, calculations for *Nπ* = 0.1, 0.3, and 0.5 were employed while Prandtl number *Pr* was kept fixed on 1.0, 6.2, and 7.38. The convective heat transfer rate rose whenever the radiation parameter *N<sup>π</sup>*, is increased. The heat transfer rate was augmented when heat flux was increased.

**Figure 2.** Velocity variation with *φ*.

**Figure 3.** Temperature variation with *φ*.

**Figure 4.** Entropy variation with *φ*.

**Figure 5.** Temperature variation with *N<sup>π</sup>*.

**Figure 6.** Entropy variation with *N<sup>π</sup>*.

**Figure 7.** Temperature variation with *υ*\*.

**Figure 8.** Entropy variation with *υ*\*.

**Figure 9.** Temperature variation with *B<sup>π</sup>*.

**Figure 10.** Entropy variation with *B<sup>π</sup>*.

**Figure 11.** Velocity variation with *τ*\*.

**Figure 12.** Temperature variation with *τ*\*.

**Figure 13.** Entropy variation with *τ*\*.

**Figure 14.** Velocity variation with *ς*\*.

**Figure 15.** Temperature variation with *ς*\*.

**Figure 16.** Entropy variation with *ς*\*.

**Figure 17.** Velocity variation with *Λπ*.

**Figure 18.** Temperature variation with *Λπ*.

**Figure 19.** Entropy variation with *Λπ*.

**Figure 20.** Skin friction *Cƒ* against the parameter *ς*\*.

**Figure 21.** Nusselt number *Nux* against the parameter *Pr*.


**Table 5.** Values of *Cf Re*1/2 *x* and *NuxRe*−1/2 *x* for *Pr* = 6450.
